Provided the mass spectrum \(m_N>m_\phi +m_\chi \) , then the FIMP DM \(\chi \)  is produced from the out-of-equilibrium decay \(N\rightarrow \chi \phi \)  [1]}. Due to mixing between the light and sterile neutrinos, the dark scalar \(\phi \)  further decays via \(\phi \rightarrow \chi \nu \) . The decay width of dark scalar \(\Gamma _\phi \simeq \lambda _{ds}^2 \theta ^2 m_\phi /8\pi \)  is heavily suppressed by the smallness of \(\lambda _{ds}\)  and \(\theta \) , therefore \(\phi \)  is a long-lived particle. For instance, with \(\lambda _{ds}\sim 10^{-11}\) , \(\theta \sim 10^{-7}\) , and \(m_\phi \sim 10^3\)  GeV, we have the lifetime \(\tau _\phi =1/\Gamma _\phi \sim 10^{10}\)  s. Such a long lifetime conflicts with constraints from CMB and BBN [2]}. One possible way to avoid the cosmological constraints is making \(\phi \)  a decaying DM via even smaller \(\lambda _{ds}\)  or \(\theta \)  [3]}. Another pathway is assuming \(m_\phi \)  lighter than 200 GeV, so that the fraction of electromagnetic energy injected into the plasma is tiny.
